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Simulating cosmic microwave background maps in multiconnected spaces
Abstract
This paper describes the computation of cosmic microwave background (CMB) anisotropies in a universe with multiconnected spatial sections and focuses on the implementation of the topology in standard CMB computer codes. The key ingredient is the computation of the eigenmodes of the Laplacian with boundary conditions compatible with multiconnected space topology. The correlators of the coefficients of the decomposition of the temperature fluctuation in spherical harmonics are computed and examples are given for spatially flat spaces and one family of spherical spaces, namely, the lens spaces. Under the hypothesis of Gaussian initial conditions, these correlators encode all the topological information of the CMB and suffice to simulate CMB maps.
... Thus, an explicit computation of the spectrum and eigenfunctions of the Laplacian on the candidate 3-manifolds, as in [23], [31], can be used to produce simulated CMB skies for the different candidate topologies, which are then compared to the WMAP data for the observed CMB. ...
... The dodecahedral space is obtained by gluing together opposite faces of a dodecahedron with the shortest clockwise twist that matches the faces. This space has been regarded as a likely candidate for the cosmic topology problem and extensively studied for testable cosmological signatures with all the methods presently available, [7], [16], [22], [24], [28], [31], [33], [39], [40]. In particular, the three-year WMAP results confirmed the main anomalies: quadrupole suppression, small value of the two-point temperature correlation function at large angles, and quadrupole–octupole alignment. ...
... Simulated CMB skies have been computed for tori and for all the Bieberbach manifolds in [32] . The method is the same as in the analysis of simulated CMB skies for spherical space forms of [23], [31], namely through the explicit computation of the spectrum and eigenforms of the Laplacian. In the case of flat tori, the basis given by planar waves is more directly adapted to the topology, while the basis in spherical waves is better suited for comparison between simulated and observed CMB sky. ...
The spectral action functional, considered as a model of gravity coupled to
matter, provides, in its non-perturbative form, a slow-roll potential for
inflation, whose form and corresponding slow-roll parameters can be sensitive
to the underlying cosmic topology. We explicitly compute the non-perturbative
spectral action for some of the main candidates for cosmic topologies, namely
the quaternionic space, the Poincare' dodecahedral space, and the flat tori. We
compute the corresponding slow-roll parameters and see we check that the
resulting inflation model behaves in the same way as for a simply-connected
spherical topology in the case of the quaternionic space and the Poincare'
homology sphere, while it behaves differently in the case of the flat tori. We
add an appendix with a discussion of the case of lens spaces.
... Instead, explanations involving a non-trivial topology of space have become popu-lar Aurich (1999a); Lehoucq et al. (1999);Inoue (1999); Levin et al. (1998);Cornish et al. (1996);de Oliveira-Costa et al. (1996); Stevens et al. (1993); Aurich (1999b); Starkman (1998); Uzan (1998);Cornish et al. (1997); Luminet & Roukema (1999); Cornish & Spergel (1999); Roukema et al. (2004); Weeks et al. (2003); Niarchou & Jaffe (2007; Kunz et al. (2008); Weeks & Gundermann (2006); Gausmann et al. (2001); Riazuelo et al. (2004b); Aurich et al. (2004); Luminet et al. (2003); Mota et al. (2004);Weeks (2003); Gomero et al. (2002a); Gomero & Rebouças (2003); Riazuelo et al. (2004a), in explaining the apparent ℓ = 2 suppression, and number of applied tests have been devised and applied to the CMB, first with the COBE data, then WMAP data, yet with somewhat unsuccessful, null only results (Cornish et al., 1998b;Roukema, 2000;de Oliveira-Costa et al., 2004;Spergel et al., 2003;Roukema et al., 2004;Phillips & Kogut, 2006;Riazuelo et al., 2006;Key et al., 2007;Caillerie et al., 2007;Lew & Roukema, 2008). Attempts were also made to unveil the shape of space via cosmic crystallography of compact spaces in different spatial curvatures by exploring the quasars and galaxy surveys and utilizing observations in frequencies from ratio to X-rays Roukema & Blanloeil (1998a); Uzan et al. (1999b); Roukema & Blanloeil (1998b); Uzan et al. (1999a); Gomero et al. (2002bGomero et al. ( , 2000; Weatherley et al. (2003), although the scales probed by these data-sets (ranging from 10 Mpc to 100 Mpc) would imply a rather small compact Universe, and naturally could not be related the to suppression detected in the CMB. ...
... Although the "matched circles" test is straightforward, it is limited due to noise and FD size constraints. Additional theoretical predictions can be used as independent tests that involve predictions of CMB temperature and polarization fluctuations for the case that the Universe is multiply connected, both in real and spherical harmonic spaces, or topological effects on the CMB power spectrum (Aurich et al., 2004;Weeks, 2003;Gomero & Rebouças, 2003;Dineen et al., 2005;Riazuelo et al., 2004a;Lehoucq et al., 1999;Inoue, 1999;Niarchou & Jaffe, 2007;Cornish et al., 1998a;de Oliveira-Costa & Smoot, 1995;Kunz et al., 2006;Luminet et al., 2003;Niarchou & Jaffe, 2006;Phillips & Kogut, 2006;Caillerie et al., 2007). Although a successful "matched circles" test would provide strong support for the Universe being multiply connected, no statistically-significant evidence has been found (Cornish et al., 2004;Key et al., 2007). ...
... The topologically interesting signal generally dominates over the Doppler (and other) components on large scales (Riazuelo et al., 2004a): this is a motivation for using a large smoothing length. However, extended flat fluctuations that happen to have a similar large scale trend can lead to false positives on large scales (Key et al., 2007). ...
名古屋大学博士学位論文 学位の種類:博士(理学) (課程) 学位授与年月日:平成20年12月31日
... The local geometry does not fix the global topology uniquely, however, and our results can be extended to various other topologies compatible with a locally spherical geometry. Signatures of a nontrivial topology in the temperature and polarization CMB spectra have been heavily studied for flat and open spatial sections (see for instance, [45,46]), and a family of spherical spaces -specifically lens and prism spaces -has been studied in [47,48] for which the mathematical and numerical groundwork was laid in [49,50]. The allowed global metrics compatible with a locally spherically symmetric geometry were classified in [51], and have the generic form S 3 /Γ, where Γ is a subgroup of isometries of S 3 acting freely and discontinuously. ...
In a recent paper [17], we studied the evolution of the background geometry and scalar perturbations in an inflationary, spatially closed Friedmann-Lema\^itre-Robertson-Walker (FLRW) model having constant positive spatial curvature and spatial topology . Due to the spatial curvature, the early phase of slow-roll inflation is modified, leading to suppression of power in the scalar power spectrum at large angular scales. In this paper, we extend the analysis to include tensor perturbations. We find that --- similarly to the scalar perturbations --- the tensor power spectrum also shows power suppression for long wavelength modes. The correction to the tensor spectrum is limited to the very long wavelength modes, therefore the resulting observable CMB B-mode polarization spectrum remains practically the same as in the standard scenario with flat spatial sections. However, since both the tensor and scalar power spectra are modified, there are scale dependent corrections to the tensor-to-scalar ratio that lead to violation of the standard slow-roll consistency relation.
... which is of great interest for cosmologists. Indeed, the non-Gaussian signatures in the CMB can be related to very fundamental questions such as the global topology of the universe [20], superstring theory, topological defects such as cosmic strings [6], and multi-field inflation [4]. The non-Gaussian signatures can, however, have a different but still cosmological origin. ...
Wavelets have been used extensively for several years now in astronomy for many purposes, ranging from data filtering and deconvolution, to star and galaxy detection or cosmic ray removal. More recent sparse representations such ridgelets or curvelets have also been proposed for the detection of anisotropic features such cosmic strings in the cosmic microwave background. We review in this paper a range of methods based on sparsity that have been proposed for astronomical data analysis. We also discuss what is the impact of Compressed Sensing, the new sampling theory, in astronomy for collecting the data, transferring them to the earth or reconstructing an image from incomplete measurements.
... We give here a brief introduction to scalar, vector and tensor harmonics on the threesphere following [20,23,24], and [25]. We write the FLRW metric as ...
We derive the power spectrum of primordial quantum fluctuations in an inflationary universe for curvature parameter K=1. This is achieved through a Born–Oppenheimer type of approximation scheme from the Wheeler–DeWitt equation of canonical quantum gravity using gauge-invariant variables. Compared to the flat model, the closed model exhibits a deficit of power at large scales.
... Hence, within the inflationary scenarios, quantum fluctuations can produce compact spaces of constant curvature, both from flat spaces [4], and curved ones [5,6]. This inspires us to think about a multi-connected finite universe, which the simplest one is the torus [7,8]. ...
We construct a cosmological model based on a free particle model which is constrained on an embedded toroidal brane, with a general rotation around a specific axis in the bulk space. Some related issues such as the rotation axis of the brane, the presence of gravitomagnetic background and its relation to the general angular velocity of the brane, and its quantum mechanics and related issues such as minimal length and minimal momentum of the quantum model in the T3 brane are studied. Also, some cosmological features such as the constraint which is imposed upon the toroidal universe by the isotropy and homogeneity conditions, the corresponding Hubble law, and accelerating expansion for the spinning toroidal model without considering a cosmological constant are also studied.
... Hence, within the inflationary scenarios, quantum fluctuations can produce compact spaces of constant curvature, both from flat spaces [4], and curved ones [5,6]. This inspires us to think about a multi-connected finite universe, which the simplest one is the torus [7,8]. ...
We construct a cosmological model based on a free particle model which is constrained on an embedded toroidal brane, with a general rotation around a specific axis in the bulk space. Some related issues such as the rotation axis of the brane, the presence of gravitomagnetic background and its relation to the general angular velocity of the brane, and its quantum mechanics and related issues such as minimal length and minimal momentum of the quantum model in the brane are studied. Also, some cosmological features such as the constraint which is imposed upon the toroidal universe by the isotropy and homogeneity conditions, the corresponding Hubble law, and accelerating expansion for the spinning toroidal model without considering a cosmological constant are also studied.
... The (countably) infinite varieties of homogeneous spherical space will always be of finite size, and all are derived from the standard three-sphere (Gausmann et al. 2001). The situation is more complicated for locally Euclidean and for hyperbolic spaces: for the former, there are 18 possibilities for the large-scale structure of space, in 10 of which space has a finite volume (Riazuelo et al. 2004); these were first classified in the context of crystallography as early as 1885! The hyperbolic case, which features an (uncountably) infinite number of spaces, some that are finite and some that are infinite in volume, is much more complicated, and the search for a complete classification an active area of mathematical research, linked to fundamental topics such as the Poincaré conjecture (Cornish, Spergel, and Starkman 1998). ...
An introduction to the physics and mathematics of the expanding universe, using no more than high-school level / undergraduate mathematics. Covered are the basics of scale factor expansion, the dynamics of the expanding universe, various distance concepts and the generalized redshift-luminosity relation, among other topics.
... The local geometry does not fix the global topology uniquely, however, and our results can be extended to various other topologies compatible with a locally spherical geometry. Signatures of a nontrivial topology in the temperature and polarization CMB spectra have been heavily studied for flat and open spatial sections (see for instance, [45,46]), and a family of spherical spaces -specifically lens and prism spaces -has been studied in [47,48] for which the mathematical and numerical groundwork was laid in [49,50]. The allowed global metrics compatible with a locally spherically symmetric geometry were classified in [51], and have the generic form S 3 /Γ, where Γ is a subgroup of isometries of S 3 acting freely and discontinuously. ...
In a recent paper [16], we studied the evolution of the background geometry and scalar perturbations in an inflationary, spatially closed Friedmann-Lema\^itre-Robertson-Walker (FLRW) model having constant positive spatial curvature and spatial topology . Due to the spatial curvature, the early phase of slow-roll inflation is modified, leading to suppression of power in the scalar power spectrum at large angular scales. In this paper, we extend the analysis to include tensor perturbations. We find that --- similarly to the scalar perturbations --- the tensor power spectrum also shows power suppression for long wavelength modes. The correction to the tensor spectrum is limited to the very long wavelength modes, therefore the resulting observable CMB B-mode polarization spectrum remains practically the same as in the standard scenario with flat spatial sections. However, since both the tensor and scalar power spectra are modified, there are scale dependent corrections to the tensor-to-scalar ratio that lead to violation of the standard slow-roll consistency relation.
... The size of the space must now reflect its curvature, linking topological properties to Ω tot = 1−Ω K . In the case of hyperbolic spacetimes, the list of possible compact spaces of constant negative curvature is still infinite, but discrete (Thurston 1982 ), while in the positive curvature spherical space there is only a finite set of well-proportioned possibilities (i.e., those with roughly comparable sizes in all directions ; there are also the denumerably infinite lens and prism topologies) for a multi-connected space (e.g., Gausmann et al. 2001; Riazuelo et al. 2004a). The effect of topology is equivalent to considering the full simply-connected three-dimensional spatial slice of the spacetime (known as the covering space) as being filled with repetitions of a shape which is finite in some or all directions (the fundamental domain) — by analogy with the two-dimensional case, we say that the fundamental domain tiles the covering space. ...
Planck CMB temperature maps allow detection of large-scale departures from
homogeneity and isotropy. We search for topology with a fundamental domain
nearly intersecting the last scattering surface (comoving distance ).
For most topologies studied the likelihood maximized over orientation shows
some preference for multi-connected models just larger than . This
effect is also present in simulated realizations of isotropic maps and we
interpret it as the alignment of mild anisotropic correlations with chance
features in a single realization; such a feature can also exist, in milder
form, when the likelihood is marginalized over orientations. Thus marginalized,
the limits on the radius of the largest sphere inscribed in a topological
domain (at log-likelihood-ratio -5) are: in a flat Universe,
for the cubic torus (cf. at 99% CL for a matched-circles
search); for the chimney; for the slab; in a
positively curved Universe, for the dodecahedron;
for the truncated cube; for the octahedron.
Similar limits apply to alternate topologies. We perform a Bayesian search for
an anisotropic Bianchi VII geometry. In a non-physical setting where the
Bianchi parameters are decoupled from cosmology, Planck data favour a Bianchi
component with a Bayes factor of at least 1.5 units of log-evidence: a Bianchi
pattern is efficient at accounting for some large-scale anomalies in Planck
data. However, the cosmological parameters are in strong disagreement with
those found from CMB anisotropy data alone. In the physically motivated setting
where the Bianchi parameters are fitted simultaneously with standard
cosmological parameters, we find no evidence for a Bianchi VII cosmology
and constrain the vorticity of such models: (95%
CL). [Abridged]
... (L is given in units of the Hubble length L H = c/H 0 ≃ 4.26Gpc for h = 0.704.) For previous works on a toroidal universe, see [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. It turns out that the torus model describes the data much better than the best-fit ΛCDM model since it exhibits the suppression of the CMB anisotropy at large scales first observed by COBE [28]. ...
We compute the effects of a compact flat universe on the angular correlation function, the angular power spectrum, the circles-in-the-sky signature, and the covariance matrix of the spherical harmonics coefficients of the cosmic microwave background radiation using the full Boltzmann physics. Our analysis shows that the Wilkinson Microwave Anisotropy Probe (WMAP) three-year data are very compatible with the possibility that we live in a flat 3-torus with volume similar or equal to 5 x 10^3 Gpc3.
... Taken at their face value, these parameters hint to a positively curved Universe. Recently Luminet et al. [35] studied the Poincaré dodecahedral space (for details, see section 2) which is one of the well-known space forms with constant positive curvature (see also [36,37,34]). The authors of ref. [35] computed the CMB multipoles for l = 2, 3 and 4, fitted the overall normalization factor to match the WMAP data at l = 4, and then examined the prediction for l = 2 and l = 3 as a function of Ω tot . ...
We analyse the anisotropy of the cosmic microwave background (CMB) for the Poincaré dodecahedron which is an example of a multi-connected spherical universe. We compare the temperature correlation function and the angular power spectrum for the Poincaré dodecahedral universe with the first-year WMAP data and find that this multi-connected universe can explain the surprisingly low CMB anisotropy on large scales found by WMAP provided that the total energy density parameter Ω_tot is in the range 1.016...1.020. The ensemble average over the primordial perturbations is assumed to be the scale-invariant Harrison–Zel'dovich spectrum. The circles-in-the-sky signature is studied and it is found that the signal of the six pairs of matched circles could be missed by current analyses of CMB sky maps.
... First, the spectrum of the Laplacian in spherical spaces is always discrete [22,23]. Second, and as emphasized in Ref. [24], a non trivial topology is most likely to be detectable in the case of spherical spaces and examples of the resulting observational effects are discussed in Ref. [25]. ...
Inter alia, the high-precision Wilkinson Microwave Anisotropy Probe (WMAP) data on cosmic background radiation marginally indicate that the Universe has positively curved (and hence spherical) spatial
sections. In this Letter, we take this data seriously and consider some of the consequences for the background dynamics. In
particular, we show that this implies a limit to the number of e-foldings that could have taken place in the inflationary
epoch; however, this limit is consistent with some inflationary models that solve all the usual cosmological problems and
that are consistent with standard structure formation theory.
... The observability of those signatures depends on the topology scale as compared to the horizon scale [206]. Detailed CMB simulations in non-trivial topologies are required to extract the maximum information from the data (see [193,100]). Precision all-sky surveys are necessary to obtain strong limits on global topology. ...
Most of the cosmological information extracted from the CMB has been obtained through the power spectrum; however, there is
much more to be learnt from the statistical distribution of the temperature random field. We review some recent developments
in the study of the cosmic microwave background (CMB) anisotropies and present a description of the novel tools developed
to analyse the properties of the CMB anisotropies beyond the power spectrum.
... which is of great interest for cosmologists. Indeed, the non-Gaussian signatures in the CMB can be related to very fundamental questions such as the global topology of the universe [20], superstring theory, topological defects such as cosmic strings [6], and multi-field inflation [4]. The non-Gaussian signatures can, however, have a different but still cosmological origin. ...
Wavelets have been used extensively for several years now in astronomy for many purposes, ranging from data filtering and deconvolution to star and galaxy detection or cosmic-ray removal. More recent sparse representations such as ridgelets or curvelets have also been proposed for the detection of anisotropic features such as cosmic strings in the cosmic microwave background. We review in this paper a range of methods based on sparsity that have been proposed for astronomical data analysis. We also discuss the impact of compressed sensing, the new sampling theory, in astronomy for collecting the data, transferring them to earth or reconstructing an image from incomplete measurements.
... For |Ω K | 1 and for 10 but for K = +1, the scalar multipole moments C s can be related to the primordial scalar curvature perturbation spectrum P ζ by [68,69,70] ...
We study a cosmological scenario in which inflation is preceded by a bounce.
In this scenario, the primordial singularity, one of the major shortcomings of
inflation, is replaced by a non-singular bounce, prior to which the universe
undergoes a phase of contraction. Our starting point is the bouncing cosmology
investigated in Falciano et al. (2008), which we complete by a detailed study
of the transfer of cosmological perturbations through the bounce and a
discussion of possible observational effects of bouncing cosmologies. We focus
on a symmetric bounce and compute the evolution of cosmological perturbations
during the contracting, bouncing and inflationary phases. We derive an
expression for the Mukhanov-Sasaki perturbation variable at the onset of the
inflationary phase that follows the bounce. Rather than being in the
Bunch-Davies vacuum, it is found to be in an excited state that depends on the
time scale of the bounce. We then show that this induces oscillations
superimposed on the nearly scale-invariant primordial spectra for scalar and
tensor perturbations. We discuss the effects of these oscillations in the
cosmic microwave background and in the matter power spectrum. We propose a new
way to indirectly measure the spatial curvature energy density parameter in the
context of this model.
... That result is for a density, and to make comparison with a mass, one multiplies of course with a volume, but which is the appropriate volume? There is the apparent volume of the presently observable universe, which takes all observational effects and their corrections into account for an expanded mass (Riazuelo et al. 2004, Lineweaver et al. 2005), but such effects did not occur for an early mass and the corrections are therefore not applicable. The comparison with the observed densities for t ~ 1 min should however take the expansion itself into account because the primordial density was observed in modern times.. ...
This paper has two parts, for a specific multiverse, and for the origin of our universe as it resulted from that multiverse. The first is based on the Planck domain and a Chandrasekhar equation that have quantum, relativity, gravity, and atomic physics in unified operation. The multiverse is an evolutionary system whereby universes survive only when they have those physics, and near-critical mass such that they do not collapse, nor expand too fast. The second part is based on 15 sets of observations of nucleosynthesis and particle properties, aging and demise for our universe, as well as of its early stages. The multiverse is supplied by debris from the aging universes, arriving on the accelerated expansion of intergalactic space. New universes accrete from the debris, which is re-energized and re-constituted gravitationally. In the process, the basic particle properties appear to have been preserved such that our universe originated much later, than it would have done in a Big Bang. The limit of Schwarzschild provides a confirmation, and information on dark energy. Comment: 20 pages
... Larger values of Ω 0 predict an unrealistically weak octopole. To obtain these predicted values we first computed the Poincaré dodecahedral space's eigenmodes using the Ghost Method of Lehoucq et al. (2002) with two of the matrix generators computed in Appendix B of Gaussmann et al. (2001), and then applied the method of Riazuelo et al. (2003), using Ω m = 0.28 and Ω Λ = Ω 0 − Ω m , to obtain a power spectrum and to simulate sky maps. Numerical limitations restricted our set of 3-dimensional eigenmodes to wavenumbers k < 30, which in turn restricted the reliable portion of the power spectrum to ℓ = 2, 3, 4. We set the overall normalisation factor to match the WMAP data at ℓ = 4 and then examined the predictions for ℓ = 2, 3. ...
The current 'standard model' of cosmology posits an infinite flat universe forever expanding under the pressure of dark energy. First-year data from the Wilkinson Microwave Anisotropy Probe (WMAP) confirm this model to spectacular precision on all but the largest scales. Temperature correlations across the microwave sky match expectations on angular scales narrower than 60 degrees but, contrary to predictions, vanish on scales wider than 60 degrees. Several explanations have been proposed. One natural approach questions the underlying geometry of space--namely, its curvature and topology. In an infinite flat space, waves from the Big Bang would fill the universe on all length scales. The observed lack of temperature correlations on scales beyond 60 degrees means that the broadest waves are missing, perhaps because space itself is not big enough to support them. Here we present a simple geometrical model of a finite space--the Poincaré dodecahedral space--which accounts for WMAP's observations with no fine-tuning required. The predicted density is Omega(0) approximately 1.013 > 1, and the model also predicts temperature correlations in matching circles on the sky.
... (L is given in units of the Hubble length L H = c/H 0 4.26 Gpc for h = 0.704.) For previous works on a toroidal universe, see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. It turns out that the torus model describes the data much better than the best-fit CDM model since it exhibits the suppression of the CMB anisotropy at large scales first observed by COBE [28]. ...
Several studies have suggested that intravenous hydroxyethyl starch treatment may dampen acute inflammatory responses. It is well documented that limited oxygen delivery to tissues (hypoxia) is common in acute inflammation, and numerous parallels exist between acute responses to hypoxia and to inflammation, including the observation that both are associated with increased vascular leakage and neutrophil infiltration of tissues. Therefore, we compared functional influences of hydroxyethyl starch on normoxic or posthypoxic endothelia.
Laboratory study.
University hospital.
Cultured human microvascular endothelial cells and mice (C57BL/6/129 svj).
We measured functional influences of hydroxyethyl starch on normoxic or posthypoxic endothelia.
Studies to assess endothelial barrier function in vitro indicated that the addition of hydroxyethyl starch promotes endothelial barrier in a dose-dependent fashion and hydroxyethyl starch-barrier effects are increased following endothelial hypoxia exposure (human microvascular endothelial cells, 48 hrs, 2% oxygen). Treatment of human microvascular endothelial cells with hydroxyethyl starch resulted in a dose-dependent increase in 157-phosphorylated vasodilator-stimulated phosphoprotein, a protein responsible for controlling the geometry of actin-filaments. Neutrophil adhesion was decreased in the presence of physiologically relevant concentrations of hydroxyethyl starch in vitro, particularly after endothelial hypoxia exposure. Using a murine model of normobaric hypoxia, increases in vascular leakage and pulmonary edema associated with hypoxia exposure (4 hrs at 8% oxygen) were decreased in animals treated with intravenous hydroxyethyl starch. Increases of tissue neutrophil accumulation following hypoxia exposure were dampened in hydroxyethyl starch-treated mice.
Taken together, these results indicate that hypoxia-induced increases in vascular leakage and acute inflammation are attenuated by hydroxyethyl starch treatment.
... A detailed description of all possible topologies of threedimensional manifolds with constant curvature were given by Wolf (1967) and Inoue (2001). A description and summary of the topologies for flat universes, which we will consider in this paper, and methods of simulations of CMB maps for such universes were also given by Riazuelo et al. (2004a) and Riazuelo et al. (2004b). We will briefly describe here, following their formalism, basic information concerning the topology of the 3-torus. ...
We study a multipole vector-based decomposition of cosmic microwave background data in order to search for signatures of a
multiconnected topology of the universe. Using 106 simulated maps, we analyse the multipole vector distribution on the sky for the lowest order multipoles together with the
probability distribution function of statistics based on the sum of the dot products of the multipole vectors for both the
simply connected flat universe and universes with the topology of a 3 torus. The estimated probabilities of obtaining lower
values for these statistics as compared to the 5-yr Wilkinson Microwave Anisotropy Probe data indicate that the observed alignment of the quadrupole and octopole is statistically favoured in a 3-torus topology
where at least one dimension of the fundamental domain is significantly shorter than the diameter of the observable Universe,
as compared to the usual standard simply connected universe. However, none of the obtained results is able to clearly rule
out the latter (at more than 97 per cent confidence level). Multipole vector statistics do not appear to be very sensitive
to the signatures of a 3-torus topology if the shorter dimension of the domain becomes comparable to the diameter of the observable
Universe. Unfortunately, the signatures are also significantly diluted by the integrated Sachs–Wolfe effect.
The curvature of a spacetime, either in a topological sense, or averaged over superhorizon-sized patches, is often equated with the global curvature term that appears in Friedmann’s equation. In general, however, the Universe is inhomogeneous, and gravity is a nonlinear theory, thus any curvature perturbations violate the assumptions of the Friedmann-Lemaïtre-Robertson-Walker model; it is not necessarily true that local curvature, averaged over patches of constant-time surfaces, will reproduce the observational effects of global symmetry. Further, the curvature of a constant-time hypersurface is not an observable quantity, and can only be inferred indirectly. Here, we examine the behavior of curvature modes on hypersurfaces of an inhomogeneous spacetime nonperturbatively in a numerical relativistic setting, and how this curvature corresponds with that inferred by observers. We also note the point at which observations become sensitive to the impact of curvature sourced by inhomogeneities on inferred average properties, finding general agreement with past literature.
This paper surveys recent work of the author and collaborators on cosmological models based on the spectral action functional of gravity. A more detailed presentation of the topics surveyed here will be available in a forthcoming book [1].
Cosmology
is the study of Nature at very large scales. Conventionally, “large scales” span a range of about 1–8000 Mpc, from our local group of galaxies to the most distant light we can possibly detect (Fig. 2.1). Phenomena occurring within a galaxy (ours or another) and associated with galactic media, stars, supernovæ, black holes, gamma-ray bursts and so on are also part of cosmology, in so far as they determine relative distances, ages, composition and gravitational properties of local patches of the universe, as well as important information on elementary particle physics and gravity.
Full-sky CMB maps from the 2015 Planck release allow us to detect departures from global isotropy on the largest scales. We present the first searches using CMB polarization for correlations induced by a non-trivial topology with a fundamental domain intersecting, or nearly intersecting, the last scattering surface (at comoving distance ). We specialize to flat spaces with toroidal and slab topologies, finding that explicit searches for the latter are sensitive to other topologies with antipodal symmetry. These searches yield no detection of a compact topology at a scale below the diameter of the last scattering surface. The limits on the radius of the largest sphere inscribed in the topological domain (at log-likelihood-ratio relative to a simply-connected flat Planck best-fit model) are for the cubic torus and for the slab. The limit for the cubic torus from the matched-circles search is numerically equivalent, (99% CL) from polarisation data alone. We also perform a Bayesian search for a Bianchi VII geometry. In the non-physical setting where the Bianchi cosmology is decoupled from the standard cosmology, Planck temperature data favour the inclusion of a Bianchi component. However, the cosmological parameters generating this pattern are in strong disagreement with those found from CMB anisotropy data alone. Fitting the induced polarization pattern for this model to Planck data requires an amplitude of compared to +1 if the model were to be correct. In the physical setting where the Bianchi parameters are fit simultaneously with the standard cosmological parameters, we find no evidence for a Bianchi VII cosmology and constrain the vorticity of such models to (95% CL). [Abridged]
The global shape, or topology, of the universe is not constrained by the equations of General Relativity, which only describe the local universe. As a consequence, the boundaries of space are not fixed and topologies different from the trivial infinite Euclidean space are possible. The cosmic microwave background (CMB) is the most efficient tool to study topology and test alternative models. Multi-connected topologies, such as the 3-torus, are of great interest because they are anisotropic and allow us to test a possible violation of isotropy in CMB data. We show that the correlation function of the coefficients of the expansion of the temperature and polarization anisotropies in spherical harmonics encodes a topological signature. This signature can be used to distinguish an infinite space from a multi-connected space on sizes larger than the diameter of the last scattering surface (D
LSS
). With the help of the Kullback-Leibler divergence, we set the size of the edge of the biggest distinguishable torus with CMB temperature fluctuations and E-modes of polarization to 1.15 D
LSS
. CMB temperature fluctuations allow us to detect universes bigger than the observable universe, whereas E-modes are efficient to detect universes smaller than the observable universe.
This article reviews the properties and limitations associated with the existence of particle, visual, and event horizons in cosmology in general and in inflationary universes in particular, carefully distinguishing them from 'Hubble horizons'. It explores to what extent one might be able to probe conditions beyond the visual horizon (which is close in size to the present Hubble radius), thereby showing that visual horizons place major limits on what are observationally testable aspects of a multiverse, if such exists. Indeed these limits largely prevent us from observationally proving a multiverse either does or does not exist. We emphasize that event horizons play no role at all in observational cosmology, even in the multiverse context, despite some claims to the contrary in the literature.
Full-sky CMB maps from the 2015 Planck release allow us to detect departures
from global isotropy on the largest scales. We present the first searches using
CMB polarization for correlations induced by a non-trivial topology with a
fundamental domain intersecting, or nearly intersecting, the last scattering
surface (at comoving distance ). We specialize to flat spaces with
toroidal and slab topologies, finding that explicit searches for the latter are
sensitive to other topologies with antipodal symmetry. These searches yield no
detection of a compact topology at a scale below the diameter of the last
scattering surface. The limits on the radius of the largest sphere
inscribed in the topological domain (at log-likelihood-ratio
relative to a simply-connected flat Planck best-fit model) are
for the cubic torus and for the slab.
The limit for the cubic torus from the matched-circles search is numerically
equivalent, (99% CL) from polarisation data alone. We also
perform a Bayesian search for a Bianchi VII geometry. In the non-physical
setting where the Bianchi cosmology is decoupled from the standard cosmology,
Planck temperature data favour the inclusion of a Bianchi component. However,
the cosmological parameters generating this pattern are in strong disagreement
with those found from CMB anisotropy data alone. Fitting the induced
polarization pattern for this model to Planck data requires an amplitude of
compared to +1 if the model were to be correct. In the physical
setting where the Bianchi parameters are fit simultaneously with the standard
cosmological parameters, we find no evidence for a Bianchi VII cosmology
and constrain the vorticity of such models to
(95% CL). [Abridged]
We illustrate the constraints that a possible detection of a non-trivial spatial topology may place on the cosmological density parameters by considering the ACDM model Poincaré dodecahedal space (PDS) topology as a circles-in-the-sky detectable topology. To this end we reanalyze the type Ia supernovae constraints on the density parameter plane Ωk - Ωλ and show that a circles-in-the-sky detectable PDS topology gives rise to important constraints on this parameters plane.
A nontrivial topology of the spatial section of the universe is an observable which can be probed for all homogeneous and isotropic universes, without any assumption on the cosmological density parameters. We discuss how one can use this observable to set constraints on the density parameters of the universe by using a specific spatial topology along with type Ia supernovae and X-ray gas mass fraction data sets.
The Hantzsche-Wendt space is one of the 17 multiply connected spaces of the
three-dimensional Euclidean space E^3. It is a compact and orientable manifold
which can serve as a model for a spatial finite universe. Since it possesses
much fewer matched back-to-back circle pairs on the cosmic microwave background
(CMB) sky than the other compact flat spaces, it can escape the detection by a
search for matched circle pairs. The suppression of temperature correlations
C(theta) on large angular scales on the CMB sky is studied. It is shown that
the large-scale correlations are of the same order as for the 3-torus topology
but express a much larger variability. The Hantzsche-Wendt manifold provides a
topological possibility with reduced large-angle correlations that can hide
from searches for matched back-to-back circle pairs.
We study the possibility for constraining the topology of the Universe by means of the matched circles statistic applied to
polarized cosmic microwave background (CMB) anisotropy maps. The advantages of using the CMB polarization maps in studies
of the topology over simply analysing the temperature data as has been done to date are clearly demonstrated. We test our
algorithm to search for pairs of matched circles on simulated CMB maps for a universe with the topology of a 3-torus. It is
found that the noise levels of both Planck and next generation CMB experiment data are no longer prohibitive and should be low enough to enable the use of the polarization
maps for such studies. For such experiments, the minimum radius of the back-to-back matched circles which can be detected
is determined. We also show that the polarization generated after reionization does not have an impact on the detectability
of the matched circles.
The standard cosmological model does not determine the spatial topology of
the universe. This article revisits the signature of a non-trivial topology on
the properties of the cosmic microwave background anisotropies. We show that
the correlation function of the coefficients of the expansion of the
temperature and polarization anisotropies in spherical harmonics, encodes a
topological signature that can be used to distinguish a multi-connected space
from an infinite space on sizes larger than the last scattering surface. The
effect of the instrumental noise and of a galactic cut are estimated. We thus
establish boundaries for the size of the biggest torus dintinguisable with
temperature and polarization CMB data. We also describe the imprint of the
spatial topology on the 3-point function and on non-Gaussianity.
Compact, flat Friedmann–Lemaître–Robertson–Walker (FLRW) models have recently regained interest as a good fit to the observed cosmic microwave background temperature fluctuations. However, it is generally thought that a globally, exactly flat FLRW model is theoretically improbable. Here, in order to obtain a probability space on the set F of compact, comoving, 3-spatial sections of FLRW models, a physically motivated hypothesis is proposed, using the density parameter Ω as a derived rather than fundamental parameter. We assume that the processes that select the 3-manifold also select a global mass-energy and a Hubble parameter. The requirement that the local and global values of Ω are equal implies a range in Ω that consists of a single real value for any 3-manifold. Thus, the obvious measure over F is the discrete measure. Hence, if the global mass-energy and Hubble parameter are a function of 3-manifold choice among compact FLRW models, then probability spaces parametrized by Ω do not, in general, give a zero probability of a flat model. Alternatively, parametrization by a spatial size parameter, the injectivity radius rinj, suggests the Lebesgue measure. In this case, the probability space over the injectivity radius implies that flat models occur almost surely (a.s.), in the sense of probability theory, and non-flat models a.s. do not occur.
What is the shape of the Universe? Is it finite or infinite ? Is space
multi-connected to create ghost images of faraway cosmic sources? After a "dark
age" period, the field of cosmic topology has now become one of the major
concerns in astronomy and cosmology, not only from theorists but also from
observational astronomers. Here I give a personal account of the spectacular
progress in the field since the beginning of the 1990's, when I started to work
in it.
In this talk I review recent advances in cosmic topology since it has entered a new era of experimental tests. High redshift surveys of astronomical sources and accurate maps of the Cosmic Microwave Background radiation
(CMB) are beginning to hint at the shape of the universe, or at least to limit the wide range of possibilities. Among those possibilites are surprising “wrap around” universe models in which space, whatever its curvature, may be smaller than the observable universe and generate topological lensing effects on a detectable cosmic scale. In particular, the recent analysis of CMB data provided by the WMAP satellite suggest a finite universe with the topology of the Poincaré dodecahedral spherical space. Such a model of a “small universe”, the volume of which would represent only about 80% the volume of the observable universe, offers an observational signature in the form of a predictable topological lens effect on one hand, and rises new issues on the early universe physics on the other hand.
We impose constraints on the topology of the Universe determined from a search for matched circles in the temperature anisotropy
patterns of the 7-yr Wilkinson Microwave Anisotropy Probe (WMAP) data. We pay special attention to the sensitivity of the method to residual foreground contamination of the sky maps and
show that for a full-sky estimate of the CMB signal (the Internal Linear Combination map) such residuals introduce a non-negligible
effect on the statistics of matched circles. In order to reduce this effect, we perform the analysis on maps for which the
most contaminated regions have been removed. A search for pairs of matched back-to-back circles in the higher resolution WMAP W-band map allows tighter constraints to be imposed on topology. Our results rule out universes with topologies that predict
pairs of such circles with radii larger than αmin≈ 10°. This places a lower bound on the size of the fundamental domain for a flat universe of about 27.9 Gpc. This bound is
close to the upper limit on the size of Universe possible to be detected by the method of matched circles, i.e. the diameter
of the observable Universe is 28.3 Gpc.
We present a new technique for constraining the topology of the Universe. The method exploits the existence of correlations
in the phases of the spherical harmonic coefficients of the cosmic microwave background (CMB) temperature pattern associated
with matched pairs of circles seen in the sky in universes with non-trivial topology. The method is computationally faster
than all other statistics developed to hunt for these matched circles. We applied the method to a range of simulations with
topologies of various forms and on different scales. A characteristic form of phase correlation is found in the simulations.
We also applied the method to preliminary CMB maps derived from WMAP, but the separation of topological effects from e.g. foregrounds is not straightforward.
This article investigates the signature of the seventeen multiconnected flat spaces in cosmic microwave background (CMB) maps. For each such space it recalls a fundamental domain and a set of generating matrices, and then goes on to find an orthonormal basis for the set of eigenmodes of the Laplace operator on that space. The basis eigenmodes are expressed as linear combinations of eigenmodes of the simply connected Euclidean space. A preceding work, which provides a general method for implementing multiconnected topologies in standard CMB codes, is then applied to simulate CMB maps and angular power spectra for each space. Unlike in the 3-torus, the results in most multiconnected flat spaces depend on the location of the observer. This effect is discussed in detail. In particular, it is shown that the correlated circles on a CMB map are generically not back to back, so that negative search of back-to-back circles in the Wilkinson Microwave Anisotropy Probe data does not exclude a vast majority of flat or nearly flat topologies.
We study how the uncertainty in the cosmological parameters impacts on the detection of topological signals, focussing on three cubic torus universes and using three tests: the information content, the S/N statistic, and the Bayesian evidence. We find, within the concordance cosmological model, that 3D torus universes with a size of ∼29 Gpc3 or larger cannot be detected. For the toroidal models that can be detected, the detection significance is primarily influenced by ΩΛ, which enters both in the noise amplitude due to the Integrated Sachs-Wolfe effect and in the size of the causal horizon which limits the accessible fundamental domain. On large angular scales ℓ<40, only ΩΛ significantly alters the detection for all three estimators considered here.
This article describes the cosmic microwave background anisotropies expected in a closed universe with the topology of a lens space L(p,q) and with a density parameter 0 close to 1. It provides the first simulated maps for such spaces along with their corresponding power spectra. In spite of our initial expectations that increasing p and thus decreasing the size of the fundamental domain should suppress the quadrupole, we find just the opposite: increasing p elevates the relative power of the low multipoles, for reasons that have since become clear. For 0 1.02, an informal ''by eye'' examination of the simulated power spectra suggests that p must be less than 15 for consistency with WMAP's data, while geometric considerations imply that matching circles will exist potentially revealing the multiconnected topology only if p7. These bounds become less stringent for values of 0 closer to 1.
We consider a model of modified gravity based on the spectral action
functional, for a cosmic topology given by a spherical space form, and the
associated slow-roll inflation scenario. We consider then the coupling of
gravity to matter determined by an almost commutative geometry over the
spherical space form. We show that this produces a multiplicative shift of the
amplitude of the power spectra for the density fluctuations and the
gravitational waves, by a multiplicative factor equal to the total number of
fermions in the matter sector of the model. We obtain the result by an explicit
nonperturbative computation, based on the Poisson summation formula and the
spectra of twisted Dirac operators on spherical space forms, as well as by a
heat-kernel computation.
I summarize various tests of general relativity on astrophysical scales, based on the large-scale structure of the universe but also on other systems, in particular the constants of physics. I emphasize the importance of hypotheses on the geometric structures of our universe while performing such tests and discuss their complementarity as well as their possible extensions.
The anisotropies of the cosmic microwave background (CMB) are computed for
the half-turn space E_2 which represents a compact flat model of the Universe,
i.e. one with finite volume. This model is inhomogeneous in the sense that the
statistical properties of the CMB depend on the position of the observer within
the fundamental cell. It is shown that the half-turn space describes the
observed CMB anisotropies on large scales better than the concordance model
with infinite volume. For most observer positions it matches the temperature
correlation function even slightly better than the well studied 3-torus
topology.
[Abridged] In a Universe with a detectable nontrivial spatial topology the last scattering surface contains pairs of matching circles with the same distribution of temperature fluctuations - the so-called circles-in-the-sky. Searches for nearly antipodal circles in maps of cosmic microwave background have so far been unsuccessful. This negative outcome along with recent theoretical results concerning the detectability of nearly flat compact topologies is sufficient to exclude a detectable nontrivial topology for most observers in very nearly flat positively and negatively curved Universes (). Here we investigate the consequences of these searches for observable nontrivial topologies if the Universe turns out to be exactly flat (). We demonstrate that in this case the conclusions deduced from such searches can be radically different. We show that for all multiply-connected orientable flat manifolds it is possible to directly study the action of the holonomies in order to obtain a general upper bound on the angle that characterizes the deviation from antipodicity of pairs of matching circles associated with the shortest closed geodesic. This bound is valid for all observers and all possible values of the compactification length parameters. We also show that in a flat Universe there are observers for whom the circles-in-the-sky searches already undertaken are insufficient to exclude the possibility of a detectable nontrivial spatial topology. It is remarkable how such small variations in the spatial curvature of the Universe, which are effectively indistinguishable geometrically, can have such a drastic effect on the detectability of cosmic topology. Comment: 6 pages. 1 Table. V2: Version to appear in Phys. Rev. D (2010). Presentation improved. Two intermediate steps corrected. Results unchanged. Typos corrected. V3: References added. Typo corrected. Published version
This text aims at discussing the relations between the cosmic acceleration and the theory of gravitation and more generally with the hypotheses underlying the construction of our cosmological model, such as the validity of general relativity on astrophysical scales and the Copernican principle. We hope to illustrate that cosmological data have now the potential of testing these hypotheses, which go beyond the measurements of its parameters. Comment: 47 Pages, 7 figures. Chapter of the book "Dark Energy: observational and theoretical approaches", Ed. by P. Ruiz-Lapuente, Cambridge University Press (2010). Version submitted on the 15 August 2008
This article provides a general study of the Hamiltonian stability and the hyperbolicity of vector field models involving both a general function of the Faraday tensor and its dual, , as well as a Proca potential for the vector field, . In particular it is demonstrated that theories involving only do not satisfy the hyperbolicity conditions. It is then shown that in this class of models, the cosmological dynamics always dilutes the vector field. In the case of a nonminimal coupling to gravity, it is established that theories involving or are generically pathologic. To finish, we exhibit a model where the vector field is not diluted during the cosmological evolution, because of a nonminimal vector field-curvature coupling which maintains second-order field equations. The relevance of such models for cosmology is discussed. Comment: 17 pages, no figure
Fermi-liquid theory (the standard model of metals) has been challenged by the discovery of anomalous properties in an increasingly large number of metals. The anomalies often occur near a quantum critical point--a continuous phase transition in the limit of absolute zero, typically between magnetically ordered and paramagnetic phases. Although not understood in detail, unusual behaviour in the vicinity of such quantum critical points was anticipated nearly three decades ago by theories going beyond the standard model. Here we report electrical resistivity measurements of the 3d metal MnSi, indicating an unexpected breakdown of the Fermi-liquid model--not in a narrow crossover region close to a quantum critical point where it is normally expected to fail, but over a wide region of the phase diagram near a first-order magnetic transition. In this regime, corrections to the Fermi-liquid model are expected to be small. The range in pressure, temperature and applied magnetic field over which we observe an anomalous temperature dependence of the electrical resistivity in MnSi is not consistent with the crossover behaviour widely seen in quantum critical systems. This may suggest the emergence of a well defined but enigmatic quantum phase of matter.
Recent released WMAP data show a low value of quadrupole in the CMB
temperature fluctuations, which confirms the early observations by COBE. In
this paper, a scenario, in which a contracting phase is followed by an
inflationary phase, is constructed. We calculate the perturbation spectrum and
show that this scenario can provide a reasonable explanation for lower CMB
anisotropies on large angular scales.
Recent observational results suggest that our universe is nearly flat and well modelled within a CDM framework. The observed values of and inevitably involve uncertainties. Motivated by this, we make a systematic study of the necessary and sufficient conditions for undetectability as well as detectability (in principle) of cosmic topology (using pattern repetition) in presence of such uncertainties. We do this by developing two complementary methods to determine detectability for nearly flat universes. Using the first method we derive analytical conditions for undetectability for infinite redshift, the accuracy of which is then confirmed by the second method. Estimates based on WMAP data together with other measurements of the density parameters are used to illustrate both methods, which are shown to provide very similar results for high redshifts. Comment: 16 pages, 1 figure, LaTeX2e
What is the shape of the Universe? Is it curved or flat, finite or infinite? Is space “wrapped around ” to create ghost images of faraway cosmic sources? We review how tessellations allow to build multiply-connected 3D Riemannian spaces useful for cosmology. We discuss more particularly the proposal of a finite, positively curved, dodecahedral space for explaining some puzzling features of the cosmic microwave background radiation, as revealed by the 2003-2006 WMAP data releases. 1 The Hall of Mirrors Imagine a room paneled with mirrors on all four vertical walls, and place ourselves somewhere within the room: a kaleidoscopic effect will be produced in the closest corner. Moreover, the repeated reflections of each pair of opposing mirrors ceaselessly reproduce the effect, creating the illusion of an infinite network extending in a plane. This paving of an infinite plane by a repeating design is called a tessellation (tessella being the name for a mosaic tile) of the Euclidean plane.
We reexamine the possibility of the detection of the cosmic topology in nearly flat hyperbolic Friedmann-Lemaître-Robertson-Walker (FLRW) universes by using patterns repetition. We update and extend our recent results in two important ways: by employing recent observational constraints on the cosmological density parameters as well as the recent mathematical results concerning small hyperbolic 3-manifolds. This produces new bounds with consequences for the detectability of the cosmic topology. In addition to obtaining new bounds, we also give a concrete example of the sensitive dependence of detectability of cosmic topology on the uncertainties in the observational values of the density parameters.
Using recent observational constraints on cosmological density parameters, together with recent mathematical results concerning small volume hyperbolic manifolds, we argue that, by employing pattern repetitions, the topology of nearly flat small hyperbolic universes can be observationally undetectable. This is important in view of the fact that quantum cosmology may favour hyperbolic universes with small volumes, and from the expectation, coming from inflationary scenarios, that Omega0 is likely to be very close to one.
This article investigates the signature of the seventeen multiconnected flat spaces in cosmic microwave background (CMB) maps. For each such space it recalls a fundamental domain and a set of generating matrices, and then goes on to find an orthonormal basis for the set of eigenmodes of the Laplace operator on that space. The basis eigenmodes are expressed as linear combinations of eigenmodes of the simply connected Euclidean space. A preceding work, which provides a general method for implementing multiconnected topologies in standard CMB codes, is then applied to simulate CMB maps and angular power spectra for each space. Unlike in the 3-torus, the results in most multiconnected flat spaces depend on the location of the observer. This effect is discussed in detail. In particular, it is shown that the correlated circles on a CMB map are generically not back to back, so that negative search of back-to-back circles in the Wilkinson Microwave Anisotropy Probe data does not exclude a vast majority of flat or nearly flat topologies.
After an introduction to the problem of cosmological structure
formation, we develop gauge invariant cosmological perturbation theory.
We derive the first order perturbation equations of Einstein's equations
and energy momentum "conservation". Furthermore, the perturbations of
Liouville's equation for collisionless particles and Boltzmann's
equation for Compton scattering are worked out. We fully discuss the
propagation of photons in a perturbed Friedmann universe, calculating
the Sachs-Wolfe effect and light deflection. The perturbation equations
are extended to accommodate also perturbations induced by seeds. With
these general results we discuss some of the main aspects of the texture
model for the formation of large scale structure in the Universe
(galaxies, clusters, sheets, voids). In this model, perturbations in the
dark matter are induced by texture seeds. The gravitational effects of a
spherically symmetric collapsing texture on dark matter, baryons and
photons are calculated in the first order perturbation theory. We study
the characteristic signature of the microwave background fluctuations
induced in this scenario and compare it with the COBE observations.
This paper investigates the computation of the eigenmodes of the Laplacian operator in multi-connected three-dimensional spherical spaces. General mathematical results and analytical solutions for lens and prism spaces are presented. Three complementary numerical methods are developed and compared with our analytic results and previous investigations. The cosmological applications of these results are discussed, focusing on the cosmic microwave background (CMB) anisotropies. In particular, whereas in the Euclidean case too-small universes are excluded by present CMB data, in the spherical case, candidate topologies will always exist even if the total energy density parameter of the universe is very close to unity.
This article describes the cosmic microwave background anisotropies expected in a closed universe with the topology of a lens space L(p,q) and with a density parameter 0 close to 1. It provides the first simulated maps for such spaces along with their corresponding power spectra. In spite of our initial expectations that increasing p and thus decreasing the size of the fundamental domain should suppress the quadrupole, we find just the opposite: increasing p elevates the relative power of the low multipoles, for reasons that have since become clear. For 0 1.02, an informal ''by eye'' examination of the simulated power spectra suggests that p must be less than 15 for consistency with WMAP's data, while geometric considerations imply that matching circles will exist potentially revealing the multiconnected topology only if p7. These bounds become less stringent for values of 0 closer to 1.
There is a correlation between the anisotropy of the background radiation and a possible topologically nontrivial nature of the universe. The discovery of a large-scale (quadrupole) anisotropy is utilized to derive a new limitation on a possible topological "radius" of the universe in the theory of an inflationary universe. This limitation is R > (2/e)RH = 0.7 RH, where RH is the radius of the observable horizon.
It is shown how a variety of periodic-orbit sum rules can be used to extract information about a quantum mechanical system, whose classical counterpart is completely chaotic, from knowledge only of the classical system, and vice versa. The basis is the Selberg trace formula, an exact analogue for the Hadamard-Gutzwiller model of the semiclassical periodic-orbit theory of Gutzwiller, which relates the quantal energies to the lengths of the periodic orbits of the classical system. Statistical properties of the quantal energies in the low-energy region are studied, where we restrict ourselves to the level spacing and spectral rigidity.
General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi- rather than simply-connected. We review the main mathematical properties of multi-connected spaces, and the different tools to classify them and to analyse their properties. Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models. We briefly discuss some implications of multi-connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe. We consider in details the properties of the cosmological models where space is multi-connected, with emphasis towards observable effects. We then review the analyses of observational results obtained in this context, to search for a possible signature of multi-connectedness, or to constrain the models. They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,…, or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them.
Statistical properties of highly excited quantal eigenstates are studied for the free motion (geodesic flow) on a compact surface of constant negative curvature (hyperbolic octagon) which represents a strongly chaotic system (K-system). The eigenstates are expanded in a circular-wave basis, and it turns out that the expansion coefficients behave as Gaussian pseudo-random numbers. It is shown that this property leads to a Gaussian amplitude distribution P(Ψ) in the semiclassical limit, i.e. the wave-functions behave as Gaussian random functions. This behaviour, which should hold for chaotic systems in general, is nicely confirmed for eigenstates lying 10 000 states above the ground state thus probing the semiclassical limit. In addition, the autocorrelation function and the path-correlation function are calculated and compared with a crude semiclassical Bessel-function approximation. Agreement with the semiclassical prediction is only found, if a local averaging is performed over roughly 1000 de Broglie wavelengths. On smaller scales, the eigenstates show much more structure than predicted by the first semiclassical approximation.
A general, gauge-invariant analysis of the large-scale anisotropies in the cosmic background radiation produced by arbitrary scalar, vector, or tensor perturbations in open, closed, or flat Robertson-Walker spacetimes (with no cosmological constant) is presented. The multipole moment predictions for the scale invariant spectrum and Omega = 1 universe predicted by inflationary cosmologies are contrasted with those for other spectra and for open and closed universes. Using the measured value of the dipole moment, limits are set on the expected value of the quadrupole moment as a function of Omega for various spectra.
The fluctuations of the cosmic microwave background (CMB) are investigated
for a hyperbolic universe with finite volume. Four-component models with
radiation, matter, vacuum energy, and an extra spatially constant dark energy
X-component are considered. The general solution of the Friedmann equation for
the cosmic scale factor a(eta) is given for the four-component models in terms
of the Weierstrass P-function. The lower part of the angular power spectra C_l
of the CMB anisotropy is computed for nearly flat models with Omega_tot <=
0.95. It is shown that the particular compact fundamental cell, which is
considered in this paper, leads to a suppression in C_l for l < 10 and Omega_tot
<= 0.9.
We revisit the observational constraints on compact(closed) hyperbolic(CH) models from cosmic microwave background(CMB). We carry out Bayesian analyses for CH models with volume comparable to the cube of the present curvature radius using the COBE-DMR data and show that a slight suppression in the large-angle temperature correlations owing to the non-trivial topology explains rather naturally the observed anomalously low quadrupole which is incompatible with the prediction of the standard infinite Friedmann-Robertson-Walker models. While most of positions and orientations are ruled out, the likelihoods of CH models are found to be much better than those of infinite counterparts for some specific positions and orientations of the observer, leading to less stringent constraints on the volume of the manifolds. Even if the spatial geometry is nearly flat as , suppression of the angular power on large angular scales is still prominent for CH models with volume much less than the cube of the present curvature radius if the cosmological constant is dominant at present. Comment: 25 pages, 16 EPS figures Version accepted for publication in PTP
In this paper, the cosmic microwave background (CMB) anisotropy in a multiply-connected compact flat 3-torus model with the cosmological constant is investigated. Using the COBE-DMR 4-year data, a full Bayesian analysis revealed that the constraint on the topology of the flat 3-torus model with low-matter-density is less stringent. As in compact hyperbolic models, the large-angle temperature fluctuations can be produced as the gravitational potential decays at the -dominant epoch well after the last scattering. The maximum allowed number N of images of the cell (fundamental domain) within the observable region at present is approximately 49 for and whereas for and . Comment: 13 pages using RevTeX, 5 eps files, typos corrected
We investigate to what extent the cosmic crystallographic methods aimed to
detect the topology of the universe using catalogues of cosmic objects would be
damaged by various observational uncertainties. We find that the topological
signature is robust in the case of Euclidean spaces, but is very fragile in the
case of compact hyperbolic spaces. Comparing our results to the presently
accepted range of values for the curvature parameters, the best hopes for
detecting space topology rest on elliptical space models.
In this paper, Gaussianity of eigenmodes and non-Gaussianity in the Cosmic Microwave Background (CMB) temperature fluctuations in two smallest compact hyperbolic (CH) models are investigated. First, it is numerically found that the expansion coefficients of low-lying eigenmodes on the two CH manifolds behave as if they are Gaussian random numbers at almost all the places. Next, non-Gaussianity of the temperature fluctuations in the (l,m) space in these models is studied. Assuming that the initial fluctuations are Gaussian, the real expansion coefficients b_{l m} of the temperature fluctuations in the sky are found to be distinctively non-Gaussian. In particular, the cosmic variances are found to be much larger than that for Gaussian models. On the other hand, the anisotropic structure is vastly erased if one averages the fluctuations at a number of different observing points because of the Gaussian pseudo-randomness of the eigenmodes. Thus the dominant contribution to the two-point correlation functions comes from the isotropic terms described by the angular power spectra C_l. Finally, topological quantities: the total length and the genus of isotemperature contours are investigated. The variances of total length and genus at high and low threshold levels are found to be considerably larger than that of Gaussian models while the means almost agree with them. Comment: 22 pages, 18 figures (eps files). Typos corrected
When the topology of the universe is non trivial, it has been shown that there are constraints on the network of domain walls, cosmic strings and monopoles. I generalize these results to textures and study the cosmological implications of such constraints. I conclude that a large class of multi-connected universes with topological defects accounting for structure formation are ruled out by observation of the cosmic microwave background.
This article gives the construction and complete classification of all three-dimensional spherical manifolds, and orders them by decreasing volume, in the context of multiconnected universe models with positive spatial curvature. It discusses which spherical topologies are likely to be detectable by crystallographic methods using three-dimensional catalogs of cosmic objects. The expected form of the pair separation histogram is predicted (including the location and height of the spikes) and is compared to computer simulations, showing that this method is stable with respect to observational uncertainties and is well suited for detecting spherical topologies.
Recent observations suggest that the ratio of the total density to the critical density of the universe, , is likely to be very close to one, with a significant proportion of this energy being in the form of a dark component with negative pressure. Motivated by this result, we study the question of observational detection of possible non-trivial topologies in universes with , which include a cosmological constant. Using a number of indicators we find that as , increasing families of possible manifolds (topologies) become either undetectable or can be excluded observationally. Furthermore, given a non-zero lower bound on , we can rule out families of topologies (manifolds) as possible candidates for the shape of our universe. We demonstrate these findings concretely by considering families of topologies and employing bounds on cosmological parameters from recent observations. We find that given the present bounds on cosmological parameters, there are families of both hyperbolic and spherical manifolds that remain undetectable and families that can be excluded as the shape of our universe. These results are of importance in future search strategies for the detection of the shape of our universe, given that there are an infinite number of theoretically possible topologies and that the future observations are expected to put a non-zero lower bound on which is more accurate and closer to zero. Comment: 18 pages, 3 figures, LaTeX2e. Two references added. Minors corrections in section 4. To appear in Class. Quantum Grav. {\bf 18} (2001) in the present form
We calculate the CMB anisotropy in compact hyperbolic universe models using the regularized method of images described in paper-I, including the 'line-of-sight `integrated Sachs-Wolfe' effect, as well as the last-scattering surface terms. We calculate the Bayesian probabilities for a selection of models by confronting our theoretical pixel-pixel temperature correlation functions with the COBE-DMR data. Our results demonstrate that strong constraints on compactness arise: if the universe is small compared to the `horizon' size, correlations appear in the maps that are irreconcilable with the observations. This conclusion is qualitatively insensitive to the matter content of the universe, in particular, the presence of a cosmological constant. If the universe is of comparable size to the 'horizon', the likelihood function is very dependent upon orientation of the manifold wrt the sky. While most orientations may be strongly ruled out, it sometimes happens that for a specific orientation the predicted correlation patterns are preferred over those for the conventional infinite models. The full Bayesian analysis we use is the most complete statistical test that can be done on the COBE maps, taking into account all possible signals and their variances in the theoretical skies, in particular the high degree of anisotropic correlation that can exist. We show that standard visual measures for comparing theoretical predictions with the data such as the isotropized power spectrum are not so useful in small compact spaces because of enhanced cosmic variance associated with the breakdown of statistical isotropy. Comment: 29 pages, Latex, 15 figures, submitted to Phys. Rev. D, March 11, 1999. Full resolution figures can be obtained from ftp://ftp.cita.utoronto.ca/pogosyan/prdB/
The measurements of CMB anisotropy have opened up a window for probing the
global topology of the universe on length scales comparable to and beyond the
Hubble radius. For compact topologies, the two main effects on the CMB are: (1)
the breaking of statistical isotropy in characteristic patterns determined by
the photon geodesic structure of the manifold and (2) an infrared cutoff in the
power spectrum of perturbations imposed by the finite spatial extent. We
present a completely general scheme using the regularized method of images for
calculating CMB anisotropy in models with nontrivial topology, and apply it to
the computationally challenging compact hyperbolic topologies. This new
technique eliminates the need for the difficult task of spatial eigenmode
decomposition on these spaces. We estimate a Bayesian probability for a
selection of models by confronting the theoretical pixel-pixel temperature
correlation function with the COBE-DMR data. Our results demonstrate that
strong constraints on compactness arise: if the universe is small compared to
the `horizon' size, correlations appear in the maps that are irreconcilable
with the observations. If the universe is of comparable size, the likelihood
function is very dependent upon orientation of the manifold wrt the sky. While
most orientations may be strongly ruled out, it sometimes happens that for a
specific orientation the predicted correlation patterns are preferred over the
conventional infinite models.
Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace-Beltrami operator. Using the direct boundary element method, we numerically obtain the low-lying eigenmodes on a compact hyperbolic 3-space called the Thurston manifold which is the second smallest in the known compact hyperbolic 3-manifolds. The computed eigenmodes are expanded in terms of eigenmodes on the unit three-dimensional pseudosphere. We numerically find that the expansion coefficients behave as Gaussian pseudo-random numbers for low-lying eigenmodes. The observed gaussianity in the CMB fluctuations can partially be attributed to the Gaussian pseudo-randomness of the expansion coefficients assuming that the Gaussian pseudo-randomness is the universal property of the compact hyperbolic spaces. Comment: 40 pages, 8 EPS figures; error estimation is included; accepted Classical and Quantum Gravity
The universe displays a three-dimensional pattern of hot and cold spots in the radiation remnant from the big bang. The global geometry of the universe can be revealed in the spatial distribution of these spots. In a topologically compact universe, distinctive patterns are especially prominent in spatial correlations of the radiation temperature. Whereas these patterns are usually washed out in statistical averages, we propose a scheme which uses the universe's spots to observe global geometry in a manner analogous to the use of multiple images of a gravitationally lensed quasar to study the geometry of the lens. To demonstrate how the geometry of space forms patterns in observations of the microwave sky, we develop a simple real-space approximation to estimate temperature correlations for any set of cosmological parameters and any global geometry. We present correlated spheres which clearly show geometric pattern formation for compact flat universes as well as for the compact negatively curved space introduced by Weeks and another discovered by Best. These examples illustrate how future satellite-based observations of the microwave background can determine the full geometry of the universe. Comment: 16 pages, 26 figures
If the universe is finite and smaller than the distance to the surface of last scatter, then the signature of the topology of the universe is writ large on the microwave background sky. We show that the microwave background will be identified at the intersections of the surface of last scattering as seen by different ``copies'' of the observer. Since the surface of last scattering is a two-sphere, these intersections will be circles, regardless of the background geometry or topology. We therefore propose a statistic that is sensitive to all small, locally homogeneous topologies. Here, small means that the distance to the surface of last scatter is smaller than the ``topology scale'' of the universe. Comment: 14 pages, 10 figures, IOP format. This paper is a direct descendant of gr-qc/9602039. To appear in a special proceedings issue of Class. Quant. Grav. covering the Cleveland Topology & Cosmology Workshop
We have refined the analysis of the data from the FIRAS (Far-InfraRed Absolute Spectrophotometer) on board the COBE (COsmic Background Explorer). The FIRAS measures the difference between the cosmic microwave background and a precise blackbody spectrum. We find new, tighter upper limits on general deviations from a blackbody spectrum. The rms deviations are less than 50 parts per million of the peak of the cosmic microwave background radiation. For the Comptonization and chemical potential, we find |y| < 15 × 10-6 and | μ | <9 × 10-5 (95% confidence level [CL]). There are also refinements in the absolute temperature, 2.728 ± 0.004 K (95% CL), the dipole direction, (ℓ, b) = (264°.14 ± 0.30, 48°.26 ±0.30) (95% CL), and the amplitude, 3.372 ±0.014 mK (95% CL). All of these results agree with our previous publications. © 1996. The American Astronomical Society. All rights reserved.
The cosmic microwave background (CMB) is a unique probe of cosmological parameters and conditions. There is a connection between anisotropy in the CMB and the topology of the Universe. Adopting a universe with the topology of a 3-Torus, or a universe where only harmonics of the fundamental mode are allowed, and using 2-years of COBE/DMR data, we obtain constraints on the topology of the Universe. Previous work constrained the topology using the slope information and the correlation function of the CMB. We obtain more accurate results by using all multipole moments, avoiding approximations by computing their full covariance matrix. We obtain the best fit for a cubic toroidal universe of scale 7200h^{-1} Mpc for n=1. The data set a lower limit on the cell size of 4320h^{-1} Mpc at 95% confidence and 5880h^{-1} Mpc at 68% confidence. These results show that the most probable cell size would be around 1.2 times larger than the horizon scale, implying that the 3-Torus topology is no longer an interesting cosmological model. Comment: Minor revisions to match published version. 14 pages, with 4 figures included. Color figures and links at http://www.sns.ias.edu/~angelica/topology.html
The linear perturbation theory of spatially homogeneous and isotropic
universes is reviewed and reformulated extensively. In the first half of
the article, a gauge-invariant formulation of the theory is carried out
with special attention paid to the geometrical meaning of the
perturbation. In the second half of the article, the application of the
theory to some important cosmological models is discussed.
If the universe has the topology of a 3-torus (T3), then it
follows from recent data on the large-scale temperature fluctuations,
ΔT/T, of the cosmic microwave background that either the minimal
size of the torus is at least on the order of the present cosmological
horizon or the large-scale ΔT/T pattern should have a symmetry
plane (in the case of the effective T1 topology), or a
symmetry axis (in the case of the effective T2 topology). The
latter possibility can probably be ruled out by the data.
This paper reviews some aspects of our knowledge of the gravitational theory of fluctuations of density in homogeneous and isotropic models of the universe. Irrotational fluid motions only are considered. All perturbations are assumed to be small and a normal mode analysis is used. The nature and time dependence of the amplitude of the various modes in expanding and contracting models of the universe are considered within the framework of Newtonian and general relativity theories. The origin of celestial structure requires that a uniform universe is unstable against arbitrarily small perturbations. However, the rate of growth of the allowed modes, particularly in an expanding universe, is sufficiently slow to cast doubt on the instability of the models.
When the topology of fit universe is non-trivial, there exist constraints on the network of topological defects that may appear during a phase transition in the early universe. Here, the author reviews, the main results of two recent papers by J. P. Uzan and P. Peter [Phys. Lett. B 406, 20-25 (1997); J.-P. Uzan, Phys. Rev. D 58 (at press) (1998)], concerning topological defects in a multiconnected universe: the no-defect conjecture, the evaluation of the redshift at which all the defects (if formed) have disappeared and the consequence on the cosmic microwave background.
ABSTRACTA widespread myth asserts that all small universe models suppress the cosmic microwave background (CMB) quadrupole. In actual fact, some models suppress the quadrupole while others elevate it, according to whether their low-order modes are weak or strong relative to their high-order modes. Elementary geometrical reasoning shows that a model's largest dimension determines the rough value ℓmin at which the CMB power spectrum ℓ(ℓ+ 1) Cℓ/2π effectively begins; for cosmologically relevant models, ℓmin≤ 3. More surprisingly, elementary geometrical reasoning shows that further reduction of a model's smaller dimensions – with its largest dimension held fixed – serves to elevate modes in the neighbourhood of ℓmin relative to the high-ℓ portion of the spectrum, rather than suppressing them as one might naively expect. Thus among the models whose largest dimension is comparable to or less than the horizon diameter, the low-order Cℓ tend to be relatively weak in well-proportioned spaces (spaces whose dimensions are approximately equal in all directions) but relatively strong in oddly proportioned spaces (spaces that are significantly longer in some directions and shorter in others). We illustrate this principle in detail for the special cases of rectangular 3-tori and spherical spaces. We conclude that well-proportioned spaces make the best candidates for a topological explanation of the low CMB quadrupole observed by COBE and WMAP.
We present in a manifestly gauge-invariant form the theory of classical linear gravitational perturbations in part I, and a quantum theory of cosmological perturbations in part II. Part I includes applications to several important examples arising in cosmology: a univese dominated by hydrodynamical matter, a universe filled with scalar-field matter, and higher-derivative theories of gravity. The growth rates of perturbations are calculated analytically in most interesting cases. The analysis is applied to study the evolution of fluctuations in inflationary universe models. Part II includes a unified description of the quantum generation and evolution of inhomogeneities about a classial Friedmann background. The method is based on standard canonical quantization of the action for cosmological perturbations which has been reduced to an expression in terms of a single gauge-invariant variable. The spectrum of density perturbations originating in quantum fluctuations is calculated in universe with hydrodynamical matter, in inflationary universe models with scalar-field matter, and in higher-derivative theories of gravity.
The gauge-invariant theory of classical and quantized cosmological perturbations developed in parts I and II is applied in part III to several interesting physical problems. It allows a simple derivation of the relation between temperature anistropes in the cosmic microwave background. radiation and the gauge-invariant potential for metric perturbations. The generation and evolution of gravitational waves is studied. As another example, a simple analysis of entropy perturbations and non-scale-invariant spectra in inflationary universe models is presented. The gauge-invariant theory of cosmological perturbations also allows a consistent and gauge-invariant definition of statistical fluctuations.
The inflationary scenario is considered, with instead of the usual The extent to which this is unnatural is quantified. On the hypothesis that the inflation field perturbation is in the conformal vacuum state, it is found that the spectrum of the primeval curvature perturbation is still given by the usual formula in terms of the inflation potential and its derivative, evaluated at the epoch when the observable universe leaves the horizon. The corresponding microwave background anisotropy is the same on small scales, and only a few times bigger on very large scales (quadrupole moment). The limit of the inflation potential coming from present limits on the anisotropy is therefore about the same.
This paper is devoted to the quantum chaology of three-dimensional systems. A
trace formula is derived for compact polyhedral billiards which tessellate the
three-dimensional hyperbolic space of constant negative curvature. The exact
trace formula is compared with Gutzwiller's semiclassical periodic-orbit theory
in three dimensions, and applied to a tetrahedral billiard being strongly
chaotic. Geometric properties as well as the conjugacy classes of the defining
group are discussed. The length spectrum and the quantal level spectrum are
numerically computed allowing the evaluation of the trace formula as is
demonstrated in the case of the spectral staircase N(E), which in turn is
successfully applied in a quantization condition.
Large-scale cosmic microwave background temperature fluctuations are calculated for a universe with the topology of a 3-torus. In such a universe only perturbations which are harmonics of the fundamental mode are permitted. By comparison with data from the Cosmic Background Explorer satellite, we find that the minimum (comoving) scale of a cubic toroidal universe is 2400/h Mpc for an n = 1 inflationary model. This is approximately an order of magnitude greater than previous limits and 80 percent of the horizon scale, implying that a topologically 'small' universe is no longer an interesting cosmological model.
We present a simple algorithm for finding eigenmodes of the Laplacian for arbitrary compact hyperbolic 3-manifolds. We apply our algorithm to a sample of twelve manifolds and generate a list of the lowest eigenvalues. We also display a selection of eigenmodes taken from the Weeks space.
It is likely that the observed distribution of the microwave background temperature over the sky is only one realization of the underlying random process associated with cosmological perturbations of quantum-mechanical origin. If so, one needs to derive the parameters of the random process, as accurately as possible, from the data of a single map. These parameters are of the utmost importance, since our knowledge of them would help us to reconstruct the dynamical evolution of the very early Universe. It appears that the lack of ergodicity of a random process on a 2-sphere does not allow us to do this with arbitrarily high accuracy. We are left with the problem of finding the best unbiased estimators of the participating parameters. A detailed solution to this problem is presented in this article. The theoretical error bars for the best unbiased estimates are derived and discussed. Comment: 26 pages, revtex; minor modifications, 8 new references, to be published in Phys. Rev. D
Nature abhors an infinity. The limits of general relativity are often signaled by infinities: infinite curvature as in the center of a black hole, the infinite energy of the singular big bang. We might be inclined to add an infinite universe to the list of intolerable infinities. Theories that move beyond general relativity naturally treat space as finite. In this review we discuss the mathematics of finite spaces and our aspirations to observe the finite extent of the universe in the cosmic background radiation. Comment: Hilarioulsy forgot to remove comments to myself in previous version. Reference added. Submitted to Physics Reports
Cosmic microwave background data shows the observable universe to be nearly flat, but leaves open the question of whether it is simply or multiply connected. Several authors have investigated whether the topology of a multiply connect hyperbolic universe would be detectable when 0.9 < Omega < 1. However, the possibility of detecting a given topology varies depending on the location of the observer within the space. Recent studies have assumed the observer sits at a favorable location. The present paper extends that work to consider observers at all points in the space, and (for given values of Omega_m and Omega_Lambda and a given topology) computes the probability that a randomly placed observer could detect the topology. The computations show that when Omega = 0.98 a randomly placed observer has a reasonable chance (~50%) of detecting a hyperbolic topology, but when Omega = 0.99 the chances are low (<10%) and decrease still further as Omega approaches one. Comment: 9 pages, 5 figures
When the density parameter is close to unity, the universe has a large curvature radius independent of its being hyperbolic or spherical, or in the limiting case of an infinite curvature radius, flat. Whatever the curvature, the universe may have either a simply connected or a multiply connected topology. In the flat case, the topology scale is arbitrary, and there is no a priori reason for this scale to be of the same order as the size of the observable universe. In the hyperbolic case, any nontrivial topology would almost surely be on a length scale too large to detect. In the spherical case, in contrast, the topology could easily occur on a detectable scale. The present paper shows how, in the spherical case, the assumption of a nearly flat universe simplifies the algorithms for detecting a multiply connected topology, but also reduces the amount of topology that can be seen. This is of primary importance for the upcoming cosmic microwave background data analysis.
This paper shows that for spherical spaces one may restrict the search to diametrically opposite pairs of circles in the circles-in-the-sky method and still detect the cyclic factor in the standard factorization of the holonomy group. This vastly decreases the algorithm's run time. If the search is widened to include pairs of candidate circles whose centres are almost opposite and whose relative twist varies slightly, then the cyclic factor along with a cyclic subgroup of the general factor may also be detected. Unfortunately, the full holonomy group is, in general, unobservable in a nearly flat spherical universe, and so a full six-parameter search is unnecessary. Crystallographic methods could also potentially detect the cyclic factor and a cyclic subgroup of the general factor, but nothing else.
We consider the most general parametrization of flat topologically compact universes, complementing the work of Scannapieco, Levin and Silk to include non-trivial shapes. We find that modifications in shape of the fundamental domain will lead to distinct signatures in the anisotropy of the cosmic microwave radiation. We make a preliminary assessment of the effect on three statistics: the angular power spectrum, the distribution of identified ``circles'' on the surface of last scattering and the correlation function of antipodal points. Comment: 5 pages, 3 figures, accepted for publication in Phys. Rev. D
We study the effect of a finite topology on the temperature correlations of the cosmic microwave background in a flat universe. Analytic expressions for the angular power spectrum are given for all possible finite flat models. We examine the angular correlation function itself, pointing out visible and discrete features that arise from topology. While observations of the power spectrum on large angular scales can be used to place bounds on the minimum topology length, cosmic variance generally restricts us from differentiating one flat topology from another. Schemes that acknowledge anisotropic structures, such as searches for ghosts, circles or geometric patterns, will be needed to further probe topology. Comment: 5 pages, 5 figures, accepted MNRAS
We generalize the total angular momentum method for computing Cosmic Microwave Background anisotropies to Friedman-Robertson-Walker (FRW) spaces with arbitrary geometries. This unifies the treatment of temperature and polarization anisotropies generated by scalar, vector and tensor perturbations of the fluid, seed, or a scalar field, in a universe with constant comoving curvature. The resulting formalism generalizes and simplifies the calculation of anisotropies and, in its integral form, allows for a fast calculation of model predictions in linear theory for any FRW metric. Comment: RevTeX 17 pgs, 2 figs included, also available at http://www.sns.ias.edu/~whu/pub.html Changes reflect published version + corrected minor typos in Eqs. (49) & (B9)
A total angular momentum representation simplifies the radiation transport problem for temperature and polarization anisotropy in the CMB. Scattering terms couple only the quadrupole moments of the distributions and each moment corresponds directly to the observable angular pattern on the sky. We develop and employ these techniques to study the general properties of anisotropy generation from scalar, vector and tensor perturbations to the metric and the matter, both in the cosmological fluids and from any seed perturbations (e.g. defects) that may be present. The simpler, more transparent form and derivation of the Boltzmann equations brings out the geometric and model-independent aspects of temperature and polarization anisotropy formation. Large angle scalar polarization provides a robust means to distinguish between isocurvature and adiabatic models for structure formation in principle. Vector modes have the unique property that the CMB polarization is dominated by magnetic type parity at small angles (a factor of 6 in power compared with 0 for the scalars and 8/13 for the tensors) and hence potentially distinguishable independent of the model for the seed. The tensor modes produce a different sign from the scalars and vectors for the temperature-polarization correlations at large angles. We explore conditions under which one perturbation type may dominate over the others including a detailed treatment of the photon-baryon fluid before recombination. Comment: 32 pg., 10 figs., RevTeX, minor changes reflect published version, minor typos corrected, also available at http://www.sns.ias.edu/~whu
With the advent of the COBE detection of fluctuations in the Cosmic Microwave Background radiation, the study of inhomogeneous cosmology has entered a new phase. It is now possible to accurately normalize fluctuations on the largest observable scales, in the linear regime. In this paper we present a model-independent method of normalizing theories to the full COBE data. This technique allows an extremely wide range of theories to be accurately normalized to COBE in a very simple and fast way. We give the best fitting normalization and relative peak likelihoods for a range of spectral shapes, and discuss the normalization for several popular theories. Additionally we present both Bayesian and frequentist measures of the goodness of fit of a representative range of theories to the COBE data. Comment: References updated, one figure redrawn
When considering perturbations in an open (Omega<1) universe, cosmologists retain only sub-curvature modes (defined as eigenfunctions of the Laplacian whose eigenvalue is less than -1 in units of the curvature scale, in contrast with the super-curvature modes whose eigenvalue is between -1 and 0). Mathematicians have known for almost half a century that all modes must be included to generate the most general HOMOGENEOUS GAUSSIAN RANDOM FIELD, despite the fact that any square integrable FUNCTION can be generated using only the sub-curvature modes. The former mathematical object, not the latter, is the relevant one for physical applications. The mathematics is here explained in a language accessible to physicists. Then it is pointed out that if the perturbations originate as a vacuum fluctuation of a scalar field there will be no super-curvature modes in nature. Finally the effect on the cmb of any super-curvature contribution is considered, which generalizes to Omega<1 the analysis given by Grishchuk and Zeldovich in 1978. A formula is given, which is used to estimate the effect. In contrast with the case Omega=1, the effect contributes to all multipoles, not just to the quadrupole. It is important to find out whether it has the same l dependence as the data, by evaluating the formula numerically. Comment: 31 pages